Serra et al. [1]} improve the bounds of the maximum number of linear regions proposed by Montufar et al. [2]} (Eq. (REF )).
Given a deep ReLU neural network with \(l\)  layers, let \(m_i\)  be the width of \(i\) -th hidden layer with \(m_i \ge 3d\) , \(d\)  is the input dimension. Serra et al. [1]} prove that the maximal number of linear regions of this neural network is lower bounded by
\(\nonumber MEC(\mathcal {N}) \ge (\prod _{i=1}^{l-1}(\lfloor \frac{m_i}{d}\rfloor +1)^{d})\sum _{j=0}^{d}\tbinom{m_l}{j}\) 
